A characterization of n-dimensional hypersurfaces in $R^{n+1}$ with commuting curvature operators

Yulian T. Tsankov. "A characterization of n-dimensional hypersurfaces in $R^{n+1}$ with commuting curvature operators." Banach Center Publications 69.1 (2005): 205-209. <http://eudml.org/doc/282030>.

@article{YulianT2005,
abstract = {Let Mⁿ be a hypersurface in $R^\{n+1\}$. We prove that two classical Jacobi curvature operators $J_x$ and $J_y$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $(K_\{x,y\} ∘ K_\{z,u\})(u) = (K_\{z,u\} ∘ K_\{x,y\})(u)$, where $K_\{x,y\}(u) = R(x,y,u)$, for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.},
author = {Yulian T. Tsankov},
journal = {Banach Center Publications},
keywords = {commutation relation; Jacobi operator; skew-symmetric operator},
language = {eng},
number = {1},
pages = {205-209},
title = {A characterization of n-dimensional hypersurfaces in $R^\{n+1\}$ with commuting curvature operators},
url = {http://eudml.org/doc/282030},
volume = {69},
year = {2005},
}

TY - JOUR
AU - Yulian T. Tsankov
TI - A characterization of n-dimensional hypersurfaces in $R^{n+1}$ with commuting curvature operators
JO - Banach Center Publications
PY - 2005
VL - 69
IS - 1
SP - 205
EP - 209
AB - Let Mⁿ be a hypersurface in $R^{n+1}$. We prove that two classical Jacobi curvature operators $J_x$ and $J_y$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $(K_{x,y} ∘ K_{z,u})(u) = (K_{z,u} ∘ K_{x,y})(u)$, where $K_{x,y}(u) = R(x,y,u)$, for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.
LA - eng
KW - commutation relation; Jacobi operator; skew-symmetric operator
UR - http://eudml.org/doc/282030
ER -